Question

Convert each angle to \(D^{\circ} M^{\prime} S^{\prime \prime}\) form. Round your answer to the nearest second. \(19.99^{\circ}\)

Short answer

19° 59' 24''

Step-by-step solution

- Identify the Whole Number of Degrees

Take the integer part of the given decimal angle. For the angle \(19.99^{\text{°}}\), the whole number of degrees is \(19^{\text{°}}\).

- Calculate the Minutes

Subtract the whole number of degrees from the given angle to find the decimal part. \(19.99^{\text{°}} - 19^{\text{°}} = 0.99^{\text{°}}\). Now, multiply this by 60 to convert it to minutes: \(0.99 \times 60 = 59.4^{\text{'}}\). As a result, we have \(19^{\text{°}} 59.4^{\text{'}}\).

- Calculate the Seconds

Take the decimal part of the minutes calculated above and multiply it by 60 to get the seconds. \(0.4 \times 60 = 24^{\text{''}}\).

- Combine and Round the Values

Combine the degrees, minutes, and seconds values: \(19^{\text{°}} 59^{\text{'}} 24^{\text{''}}\). Since the seconds value is already a whole number, no rounding is necessary.

Key concepts

decimal degrees

Decimal degrees is a way to express angles using decimal numbers instead of the traditional sexagesimal notation, which uses degrees, minutes, and seconds.
This method is particularly useful in fields like cartography and astronomy where precision is essential.
For example, an angle of 19.99° can be more useful for calculations than an angle in degrees, minutes, and seconds.
It's quite simple to understand. Imagine dividing a single degree into 100 parts. Each part would be represented as 0.01°. So, 19.99° means 19 full degrees and almost another full degree.

degrees minutes seconds

Degrees, minutes, and seconds (DMS) is an older, yet still widely used system for expressing angles. One degree is made up of 60 minutes, and each minute is made up of 60 seconds. This system was developed a long time ago and reflects the way we often divide time and angles.
When converting a decimal degree to DMS, you start by taking the whole number of degrees. For 19.99°, that would be 19°. The remaining decimal part (0.99°) is converted to minutes by multiplying by 60, resulting in 59.4'.
Minutes are further broken down into seconds by multiplying the decimal part of the minutes by 60, resulting in 24''. So, 19.99° becomes 19° 59' 24''.
This conversion helps with precision as it allows angles to be broken down into smaller, more manageable units.

trigonometry

Trigonometry is a branch of mathematics that studies the relationships between the sides and angles of triangles. It plays a crucial role in various scientific and engineering fields.
Understanding angle measurements, such as decimal degrees and DMS, is fundamental in trigonometry because trigonometric functions, like sine, cosine, and tangent, rely on angle inputs.
Whether you're working on a physics problem, calculating distances in land surveying, or designing structures, these angle conversions will come in handy. Knowing how to switch between decimal degrees and DMS makes it easier to apply trigonometric principles in real-world scenarios.
It's not just about solving exercises; it's about gaining the skills to apply this knowledge practically.